# Monte-Carlo Simulations and Analysis of Stochastic Differential Equations" In Sim.DiffProc: Simulation of Diffusion Processes

library(Sim.DiffProc)
library(knitr)
knitr::opts_chunk$set(comment="",prompt=TRUE, fig.show='hold', warning=FALSE, message=FALSE) options(prompt="R> ",scipen=16,digits=5,warning=FALSE, message=FALSE, width = 70)  # snssde1d() Assume that we want to describe the following SDE: Ito form^[The equivalently of$X_{t}^{\text{mod1}}$the following Stratonovich SDE:$dX_{t} = \theta X_{t} \circ dW_{t}$.]: $$\label{eq:05} dX_{t} = \frac{1}{2}\theta^{2} X_{t} dt + \theta X_{t} dW_{t},\qquad X_{0}=x_{0} > 0$$ Stratonovich form: $$\label{eq:06} dX_{t} = \frac{1}{2}\theta^{2} X_{t} dt +\theta X_{t} \circ dW_{t},\qquad X_{0}=x_{0} > 0$$ In the above$f(t,x)=\frac{1}{2}\theta^{2} x$and$g(t,x)= \theta x$($\theta > 0$),$W_{t}$is a standard Wiener process. To simulate this models using snssde1d() function we need to specify: • The drift and diffusion coefficients as R expressions that depend on the state variable x and time variable t. • The number of simulation steps N=1000 (by default: N=1000). • The number of the solution trajectories to be simulated by M=1000 (by default: M=1). • The initial conditions t0=0, x0=10 and end time T=1 (by default: t0=0, x0=0 and T=1). • The integration step size Dt=0.001 (by default: Dt=(T-t0)/N). • The choice of process types by the argument type="ito" for Ito or type="str" for Stratonovich (by default type="ito"). • The numerical method to be used by method (by default method="euler"). set.seed(1234, kind = "L'Ecuyer-CMRG") theta = 0.5 f <- expression( (0.5*theta^2*x) ) g <- expression( theta*x ) mod1 <- snssde1d(drift=f,diffusion=g,x0=10,M=1000,type="ito") # Using Ito mod2 <- snssde1d(drift=f,diffusion=g,x0=10,M=1000,type="str") # Using Stratonovich mod1 mod2  Using Monte-Carlo simulations, the following statistical measures (S3 method) for class snssde1d() can be approximated for the$X_{t}$process at any time$t$: • The expected value$\text{E}(X_{t})$at time$t$, using the command mean. • The variance$\text{Var}(X_{t})$at time$t$, using the command moment with order=2 and center=TRUE. • The median$\text{Med}(X_{t})$at time$t$, using the command Median. • The mode$\text{Mod}(X_{t})$at time$t$, using the command Mode. • The quartile of$X_{t}$at time$t$, using the command quantile. • The maximum and minimum of$X_{t}$at time$t$, using the command min and max. • The skewness and the kurtosis of$X_{t}$at time$t$, using the command skewness and kurtosis. • The coefficient of variation (relative variability) of$X_{t}$at time$t$, using the command cv. • The central moments up to order$p$of$X_{t}$at time$t$, using the command moment. • The empirical$\alpha \%$confidence interval of expected value$\text{E}(X_{t})$at time$t$(from the$2.5th$to the$97.5th$percentile), using the command bconfint. • The result summaries of the results of Monte-Carlo simulation at time$t$, using the command summary. The summary of the results of mod1 and mod2 at time$t=1$of class snssde1d() is given by: summary(mod1, at = 1) summary(mod2, at = 1)  Hence we can just make use of the rsde1d() function to build our random number generator for the conditional density of the$X_{t}|X_{0}$($X_{t}^{\text{mod1}}| X_{0}$and$X_{t}^{\text{mod2}}|X_{0}$) at time$t = 1$. x1 <- rsde1d(object = mod1, at = 1) # X(t=1) | X(0)=x0 (Ito SDE) x2 <- rsde1d(object = mod2, at = 1) # X(t=1) | X(0)=x0 (Stratonovich SDE) head(data.frame(x1,x2),n=5)  The function dsde1d() can be used to show the Approximate transitional density for$X_{t}|X_{0}$at time$t-s=1$with log-normal curves: mu1 = log(10); sigma1= sqrt(theta^2) # log mean and log variance for mod1 mu2 = log(10)-0.5*theta^2 ; sigma2 = sqrt(theta^2) # log mean and log variance for mod2 AppdensI <- dsde1d(mod1, at = 1) AppdensS <- dsde1d(mod2, at = 1) plot(AppdensI , dens = function(x) dlnorm(x,meanlog=mu1,sdlog = sigma1)) plot(AppdensS , dens = function(x) dlnorm(x,meanlog=mu2,sdlog = sigma2))  r|X_{0}$ at time $t-s=1$ with log-normal curves. mod1: Ito and mod2: Stratonovich ', fig.env='figure*',fig.width=10,fig.height=10} knitr::include_graphics(c("Figures/fig007.png","Figures/fig008.png"))

In Figure 2, we present the flow of trajectories, the mean path (red lines) of solution of \eqref{eq:05} and \eqref{eq:06}, with their empirical $95\%$ confidence bands, that is to say from the $2.5th$ to the $97.5th$ percentile for each observation at time $t$ (blue lines):

r
## Ito
plot(mod1,ylab=expression(X^mod1))
lines(time(mod1),apply(mod1$X,1,mean),col=2,lwd=2) lines(time(mod1),apply(mod1$X,1,bconfint,level=0.95)[1,],col=4,lwd=2)
lines(time(mod1),apply(mod1$X,1,bconfint,level=0.95)[2,],col=4,lwd=2) legend("topleft",c("mean path",paste("bound of", 95,"% confidence")),inset = .01,col=c(2,4),lwd=2,cex=0.8) ## Stratonovich plot(mod2,ylab=expression(X^mod2)) lines(time(mod2),apply(mod2$X,1,mean),col=2,lwd=2)
lines(time(mod2),apply(mod2$X,1,bconfint,level=0.95)[1,],col=4,lwd=2) lines(time(mod2),apply(mod2$X,1,bconfint,level=0.95)[2,],col=4,lwd=2)
legend("topleft",c("mean path",paste("bound of",95,"% confidence")),col=c(2,4),inset =.01,lwd=2,cex=0.8)

knitr::include_graphics(c("Figures/fig07.png","Figures/fig08.png"))


# snssde2d()

The following $2$-dimensional SDE's with a vector of drift and matrix of diffusion coefficients:

Ito form: $$\label{eq:09} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t}) dt + g_{x}(t,X_{t},Y_{t}) dW_{1,t}\ dY_t = f_{y}(t,X_{t},Y_{t}) dt + g_{y}(t,X_{t},Y_{t}) dW_{2,t} \end{cases}$$

Stratonovich form: $$\label{eq:10} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t}) dt + g_{x}(t,X_{t},Y_{t}) \circ dW_{1,t}\ dY_t = f_{y}(t,X_{t},Y_{t}) dt + g_{y}(t,X_{t},Y_{t}) \circ dW_{2,t} \end{cases}$$ where $(W_{1,t}, W_{2,t})$ are a two independent standard Wiener process if corr = NULL. To simulate $2d$ models using snssde2d() function we need to specify:

• The drift (2d) and diffusion (2d) coefficients as R expressions that depend on the state variable x, y and time variable t.
• corr the correlation structure of two standard Wiener process $(W_{1,t},W_{2,t})$; must be a real symmetric positive-definite square matrix of dimension $2$ (default: corr=NULL).
• The number of simulation steps N (default: N=1000).
• The number of the solution trajectories to be simulated by M (default: M=1).
• The initial conditions t0, x0 and end time T (default: t0=0, x0=c(0,0) and T=1).
• The integration step size Dt (default: Dt=(T-t0)/N).
• The choice of process types by the argument type="ito" for Ito or type="str" for Stratonovich (default type="ito").
• The numerical method to be used by method (default method="euler").

## Ornstein-Uhlenbeck process and its integral

The Ornstein-Uhlenbeck (OU) process has a long history in physics. Introduced in essence by Langevin in his famous 1908 paper on Brownian motion, the process received a more thorough mathematical examination several decades later by Uhlenbeck and Ornstein (1930). The OU process is understood here to be the univariate continuous Markov process $X_t$. In mathematical terms, the equation is written as an Ito equation: $$\label{eq016} dX_t = -\frac{1}{\mu} X_t dt + \sqrt{\sigma} dW_t,\quad X_{0}=x_{0}$$ In these equations, $\mu$ and $\sigma$ are positive constants called, respectively, the relaxation time and the diffusion constant. The time integral of the OU process $X_t$ (or indeed of any process $X_t$) is defined to be the process $Y_t$ that satisfies: $$\label{eq017} Y_{t} = Y_{0}+\int X_{t} dt \Leftrightarrow dY_t = X_{t} dt ,\quad Y_{0}=y_{0}$$ $Y_t$ is not itself a Markov process; however, $X_t$ and $Y_t$ together comprise a bivariate continuous Markov process. We wish to find the solutions $X_t$ and $Y_t$ to the coupled time-evolution equations: $$\label{eq018} \begin{cases} dX_t = -\frac{1}{\mu} X_t dt + \sqrt{\sigma} dW_t\ dY_t = X_{t} dt \end{cases}$$

We simulate a flow of $1000$ trajectories of $(X_{t},Y_{t})$, with integration step size $\Delta t = 0.01$, and using second Milstein method.

set.seed(1234, kind = "L'Ecuyer-CMRG")
x0=5;y0=0
mu=3;sigma=0.5
fx <- expression(-(x/mu),x)
gx <- expression(sqrt(sigma),0)
mod2d <- snssde2d(drift=fx,diffusion=gx,Dt=0.01,M=1000,x0=c(x0,y0),method="smilstein")
mod2d


The summary of the results of mod2d at time $t=10$ of class snssde2d() is given by:

summary(mod2d, at = 10)


For plotting in time (or in plane) using the command plot (plot2d), the results of the simulation are shown in Figure 3.

## in time
plot(mod2d)
## in plane (O,X,Y)
plot2d(mod2d,type="n")
points2d(mod2d,col=rgb(0,100,0,50,maxColorValue=255), pch=16)

knitr::include_graphics(c("Figures/fig09.png","Figures/fig009.png"))


Hence we can just make use of the rsde2d() function to build our random number for $(X_{t},Y_{t})$ at time $t = 10$.

out <- rsde2d(object = mod2d, at = 10)


The density of $X_t$ and $Y_t$ at time $t=10$ are reported using dsde2d() function, see e.g. Figure 4: the marginal density of $X_t$ and $Y_t$ at time $t=10$. For plotted in (x, y)-space with dim = 2. A contour and image plot of density obtained from a realization of system $(X_{t},Y_{t})$ at time t=10, see:

## the marginal density
denM <- dsde2d(mod2d,pdf="M",at =10)
plot(denM, main="Marginal Density")
## the Joint density
denJ <- dsde2d(mod2d, pdf="J", n=100,at =10)
plot(denJ,display="contour",main="Bivariate Transition Density at time t=10")

knitr::include_graphics(c("Figures/fig1001.png","Figures/fig1002.png"))


A $3$D plot of the transition density at $t=10$ obtained with:

plot(denJ,display="persp",main="Bivariate Transition Density at time t=10")

knitr::include_graphics(c("Figures/fig1003.png"))


We approximate the bivariate transition density over the set transition horizons $t\in [1,10]$ by $\Delta t = 0.005$ using the code:

for (i in seq(1,10,by=0.005)){
plot(dsde2d(mod2d, at = i,n=100),display="contour",main=paste0('Transition Density \n t = ',i))
}


## The stochastic Van-der-Pol equation

The Van der Pol (1922) equation is an ordinary differential equation that can be derived from the Rayleigh differential equation by differentiating and setting $\dot{x}=y$, see Naess and Hegstad (1994); Leung (1995) and for more complex dynamics in Van-der-Pol equation see Jing et al. (2006). It is an equation describing self-sustaining oscillations in which energy is fed into small oscillations and removed from large oscillations. This equation arises in the study of circuits containing vacuum tubes and is given by: $$\label{eq:12} \ddot{X}-\mu (1-X^{2}) \dot{X} + X = 0$$ where $x$ is the position coordinate (which is a function of the time $t$), and $\mu$ is a scalar parameter indicating the nonlinearity and the strength of the damping, to simulate the deterministic equation see Grayling (2014) for more details. Consider stochastic perturbations of the Van-der-Pol equation, and random excitation force of such systems by White noise $\xi_{t}$, with delta-type correlation function $\text{E}(\xi_{t}\xi_{t+h})=2\sigma \delta (h)$ $$\label{eq:13} \ddot{X}-\mu (1-X^{2}) \dot{X} + X = \xi_{t},$$ where $\mu > 0$ . It's solution cannot be obtained in terms of elementary functions, even in the phase plane. The White noise $\xi_{t}$ is formally derivative of the Wiener process $W_{t}$. The representation of a system of two first order equations follows the same idea as in the deterministic case by letting $\dot{x}=y$, from physical equation we get the above system: $$\label{eq:14} \begin{cases} \dot{X} = Y \ \dot{Y} = \mu \left(1-X^{2}\right) Y - X + \xi_{t} \end{cases}$$ The system can mathematically explain by a Stratonovitch equations: \label{eq:15} \begin{cases} dX_{t} = Y_{t} dt \ dY_{t} = \left(\mu (1-X^{2}{t}) Y{t} - X_{t}\right) dt + 2 \sigma \circ dW_{2,t} \end{cases}

Implemente in R as follows, with integration step size $\Delta t = 0.01$ and using stochastic Runge-Kutta methods 1-stage.

set.seed(1234, kind = "L'Ecuyer-CMRG")
mu = 4; sigma=0.1
fx <- expression( y ,  (mu*( 1-x^2 )* y - x))
gx <- expression( 0 ,2*sigma)
mod2d <- snssde2d(drift=fx,diffusion=gx,N=10000,Dt=0.01,type="str",method="rk1")


For plotting (back in time) using the command plot, and plot2d in plane the results of the simulation are shown in Figure 6.

plot(mod2d,ylim=c(-8,8))   ## back in time
plot2d(mod2d)              ## in plane (O,X,Y)

knitr::include_graphics(c("Figures/fig1004.png","Figures/fig1005.png"))


## The Heston Model

Consider a system of stochastic differential equations:

$$\label{eq:115} \begin{cases} dX_{t} = \mu X_{t} dt + X_{t}\sqrt{Y_{t}} dB_{1,t}\ dY_{t} = \nu (\theta-Y_{t}) dt + \sigma \sqrt{Y_{t}} dB_{2,t} \end{cases}$$

Conditions to ensure positiveness of the volatility process are that $2\nu \theta > \sigma^2$, and the two Brownian motions $(B_{1,t},B_{2,t})$ are correlated. $\Sigma$ to describe the correlation structure, for example: $$\Sigma= \begin{pmatrix} 1 & 0.3 \ 0.3 & 2 \end{pmatrix}$$

set.seed(1234, kind = "L'Ecuyer-CMRG")
mu = 1.2; sigma=0.1;nu=2;theta=0.5
fx <- expression( mu*x ,nu*(theta-y))
gx <- expression( x*sqrt(y) ,sigma*sqrt(y))
Sigma <- matrix(c(1,0.3,0.3,2),nrow=2,ncol=2) # correlation matrix
HM <- snssde2d(drift=fx,diffusion=gx,Dt=0.001,x0=c(100,1),corr=Sigma,M=1000)
HM


Hence we can just make use of the rsde2d() function to build our random number for $(X_{t},Y_{t})$ at time $t = 1$.

out <- rsde2d(object = HM, at = 1)


The density of $X_t$ and $Y_t$ at time $t=1$ are reported using dsde2d() function. See:

denJ <- dsde2d(HM,pdf="J",at =1,lims=c(-100,900,0.4,0.75))
plot(denJ,display="contour",main="Bivariate Transition Density at time t=10")
plot(denJ,display="persp",main="Bivariate Transition Density at time t=10")


# snssde3d()

The following $3$-dimensional SDE's with a vector of drift and matrix of diffusion coefficients:

Ito form: $$\label{eq17} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t},Z_{t}) dt + g_{x}(t,X_{t},Y_{t},Z_{t}) dW_{1,t}\ dY_t = f_{y}(t,X_{t},Y_{t},Z_{t}) dt + g_{y}(t,X_{t},Y_{t},Z_{t}) dW_{2,t}\ dZ_t = f_{z}(t,X_{t},Y_{t},Z_{t}) dt + g_{z}(t,X_{t},Y_{t},Z_{t}) dW_{3,t} \end{cases}$$

Stratonovich form: $$\label{eq18} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t},Z_{t}) dt + g_{x}(t,X_{t},Y_{t},Z_{t}) \circ dW_{1,t}\ dY_t = f_{y}(t,X_{t},Y_{t},Z_{t}) dt + g_{y}(t,X_{t},Y_{t},Z_{t}) \circ dW_{2,t}\ dZ_t = f_{z}(t,X_{t},Y_{t},Z_{t}) dt + g_{z}(t,X_{t},Y_{t},Z_{t}) \circ dW_{3,t} \end{cases}$$ $(W_{1,t},W_{2,t},W_{3,t})$ are three independents standard Wiener process if corr = NULL. To simulate this system using snssde3d() function we need to specify:

• The drift (3d) and diffusion (3d) coefficients as R expressions that depend on the state variables x, y , z and time variable t.
• corr the correlation structure of three standard Wiener process $(W_{1,t},W_{2,t},W_{2,t})$; must be a real symmetric positive-definite square matrix of dimension $3$ (default: corr=NULL).
• The number of simulation steps N (default: N=1000).
• The number of the solution trajectories to be simulated by M (default: M=1).
• The initial conditions t0, x0 and end time T (default: t0=0, x0=c(0,0,0) and T=1).
• The integration step size Dt (default: Dt=(T-t0)/N).
• The choice of process types by the argument type="ito" for Ito or type="str" for Stratonovich (default type="ito").
• The numerical method to be used by method (default method="euler").

## Basic example

Assume that we want to describe the following SDE's (3D) in Ito form: $$\label{eq0166} \begin{cases} dX_t = 4 (-1-X_{t}) Y_{t} dt + 0.2 dW_{1,t}\ dY_t = 4 (1-Y_{t}) X_{t} dt + 0.2 dW_{2,t}\ dZ_t = 4 (1-Z_{t}) Y_{t} dt + 0.2 dW_{3,t} \end{cases}$$ with $(W_{1,t},W_{2,t},W_{3,t})$ are three indpendant standard Wiener process.

We simulate a flow of $1000$ trajectories, with integration step size $\Delta t = 0.001$.

set.seed(1234, kind = "L'Ecuyer-CMRG")
fx <- expression(4*(-1-x)*y , 4*(1-y)*x , 4*(1-z)*y)
gx <- rep(expression(0.2),3)
mod3d <- snssde3d(x0=c(x=2,y=-2,z=-2),drift=fx,diffusion=gx,M=1000)
mod3d


The following statistical measures (S3 method) for class snssde3d() can be approximated for the $(X_{t},Y_{t},Z_{t})$ process at any time $t$, for example at=1:

s = 1
mean(mod3d, at = s)
moment(mod3d, at = s , center = TRUE , order = 2) ## variance
Median(mod3d, at = s)
Mode(mod3d, at = s)
quantile(mod3d , at = s)
kurtosis(mod3d , at = s)
skewness(mod3d , at = s)
cv(mod3d , at = s )
min(mod3d , at = s)
max(mod3d , at = s)
moment(mod3d, at = s , center= TRUE , order = 4)
moment(mod3d, at = s , center= FALSE , order = 4)


The summary of the results of mod3d at time $t=1$ of class snssde3d() is given by:

summary(mod3d, at = t)


For plotting (back in time) using the command plot, and plot3D in space the results of the simulation are shown in Figure 7.

plot(mod3d,union = TRUE)         ## back in time
plot3D(mod3d,display="persp")    ## in space (O,X,Y,Z)

knitr::include_graphics(c("Figures/fig10.png","Figures/fig11.png"))


Hence we can just make use of the rsde3d() function to build our random number for $(X_{t},Y_{t},Z_{t})$ at time $t = 1$.

out <- rsde3d(object = mod3d, at = 1)


For each SDE type and for each numerical scheme, the marginal density of $X_t$, $Y_t$ and $Z_t$ at time $t=1$ are reported using dsde3d() function, see e.g. Figure 8.

den <- dsde3d(mod3d,pdf="M",at =1)
plot(den, main="Marginal Density")


For an approximate joint transition density for $(X_t,Y_t,Z_t)$ (for more details, see package sm or ks.)

denJ <- dsde3d(mod3d,pdf="J")
plot(denJ,display="rgl")


## Attractive model for 3D diffusion processes

If we assume that $U_w( x , y , z , t )$, $V_w( x , y , z , t )$ and $S_w( x , y , z , t )$ are neglected and the dispersion coefficient $D( x , y , z )$ is constant. A system becomes (see Boukhetala,1996): \begin{eqnarray}\label{eq19} % \nonumber to remove numbering (before each equation) \begin{cases} dX_t = \left(\frac{-K X_{t}}{\sqrt{X^{2}{t} + Y^{2}{t} + Z^{2}{t}}}\right) dt + \sigma dW{1,t} \nonumber\ dY_t = \left(\frac{-K Y_{t}}{\sqrt{X^{2}{t} + Y^{2}{t} + Z^{2}{t}}}\right) dt + \sigma dW{2,t} \ dZ_t = \left(\frac{-K Z_{t}}{\sqrt{X^{2}{t} + Y^{2}{t} + Z^{2}{t}}}\right) dt + \sigma dW{3,t} \nonumber \end{cases} \end{eqnarray} with initial conditions $(X_{0},Y_{0},Z_{0})=(1,1,1)$, by specifying the drift and diffusion coefficients of three processes $X_{t}$, $Y_{t}$ and $Z_{t}$ as R expressions which depends on the three state variables (x,y,z) and time variable t, with integration step size Dt=0.0001.

set.seed(1234, kind = "L'Ecuyer-CMRG")
K = 4; s = 1; sigma = 0.2
fx <- expression( (-K*x/sqrt(x^2+y^2+z^2)) , (-K*y/sqrt(x^2+y^2+z^2)) , (-K*z/sqrt(x^2+y^2+z^2)) )
gx <- rep(expression(sigma),3)
mod3d <- snssde3d(drift=fx,diffusion=gx,N=10000,x0=c(x=1,y=1,z=1))


The results of simulation (3D) are shown in Figure 9:

plot3D(mod3d,display="persp",col="blue")


## Transformation of an SDE one-dimensional

Next is an example of one-dimensional SDE driven by three correlated Wiener process ($B_{1,t}$,$B_{2,t}$,$B_{3,t}$), as follows: $$\label{eq20} dX_{t} = B_{1,t} dt + B_{2,t} dB_{3,t}$$ with: $$\Sigma= \begin{pmatrix} 1 & 0.2 &0.5\ 0.2 & 1 & -0.7 \ 0.5 &-0.7&1 \end{pmatrix}$$ To simulate the solution of the process $X_t$, we make a transformation to a system of three equations as follows: \begin{eqnarray}\label{eq21} \begin{cases} % \nonumber to remove numbering (before each equation) dX_t = Y_{t} dt + Z_{t} dB_{3,t} \nonumber\ dY_t = dB_{1,t} \ dZ_t = dB_{2,t} \nonumber \end{cases} \end{eqnarray} run by calling the function snssde3d() to produce a simulation of the solution, with $\mu = 1$ and $\sigma = 1$.

set.seed(1234, kind = "L'Ecuyer-CMRG")
fx <- expression(y,0,0)
gx <- expression(z,1,1)
Sigma <-matrix(c(1,0.2,0.5,0.2,1,-0.7,0.5,-0.7,1),nrow=3,ncol=3)
modtra <- snssde3d(drift=fx,diffusion=gx,M=1000,corr=Sigma)
modtra


The histogram and kernel density of $X_t$ at time $t=1$ are reported using rsde3d() function, and we calculate emprical variance-covariance matrix $C(s,t)=\text{Cov}(X_{s},X_{t})$, see e.g. Figure 10.

X <- rsde3d(modtra,at=1)$x MASS::truehist(X,xlab = expression(X[t==1]));box() lines(density(X),col="red",lwd=2) legend("topleft",c("Distribution histogram","Kernel Density"),inset =.01,pch=c(15,NA),lty=c(NA,1),col=c("cyan","red"), lwd=2,cex=0.8) ## Cov-Matrix color.palette=colorRampPalette(c('white','green','blue','red')) filled.contour(time(modtra), time(modtra), cov(t(modtra$X)), color.palette=color.palette,plot.title = title(main = expression(paste("Covariance empirique:",cov(X[s],X[t]))),xlab = "time", ylab = "time"),key.title = title(main = ""))

knitr::include_graphics(c("Figures/fig1007.png","Figures/fig1006.png"))


# References

1. Boukhetala K (1996). Modelling and Simulation of a Dispersion Pollutant with Attractive Centre, volume 3, pp. 245-252. Computer Methods and Water Resources, Computational Mechanics Publications, Boston, USA.

2. Guidoum AC, Boukhetala K (2020). "Performing Parallel Monte Carlo and Moment Equations Methods for ItÃ´ and Stratonovich Stochastic Differential Systems: R Package Sim.DiffProc". Journal of Statistical Software, 96(2), 1--82. https://doi.org/10.18637/jss.v096.i02

## Try the Sim.DiffProc package in your browser

Any scripts or data that you put into this service are public.

Sim.DiffProc documentation built on May 29, 2024, 8:09 a.m.